Problem: Factor the following expression: $-3$ $x^2+$ $8$ $x+$ $35$
Answer: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-3)}{(35)} &=& -105 \\ {a} + {b} &=& & & {8} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-105$ and add them together. Remember, since $-105$ is negative, one of the factors must be negative. The factors that add up to ${8}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-7}$ and ${b}$ is ${15}$ $ \begin{eqnarray} {ab} &=& ({-7})({15}) &=& -105 \\ {a} + {b} &=& {-7} + {15} &=& 8 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-3}x^2 {-7}x +{15}x +{35} $ Group the terms so that there is a common factor in each group: $ ({-3}x^2 {-7}x) + ({15}x +{35}) $ Factor out the common factors: $ x(-3x - 7) - 5(-3x - 7) $ Notice how $(-3x - 7)$ has become a common factor. Factor this out to find the answer. $(-3x - 7)(x - 5)$